The main result of this paper is using Bishop Frame and “Type-2 Bishop Frame” to study the cusps of Bishop spherical images and type-2 Bishop spherical images which are deeply related to a space curve and to make them visualized by computer. We find that the singular points of the Bishop spherical images and type-2 Bishop spherical images correspond to the point where Bishop curvatures and type-2 Bishop curvatures vanished and their derivatives are not equal to zero. As applications and illustration of the main results, two examples are given.

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E 1 , 1 , g λ 1 , λ 2 , where λ 1 ≥ λ 2 > 0 . It provides a natural 2 -parametric deformation family of the Riemannian homogeneous manifold Sol 3 = E 1 , 1 , g 1 , 1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in E 1 , 1 , g L λ 1 , λ 2 away from characteristic points and signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.

The universal covering group of Euclidean motion group E(2) with the general left-invariant metric is denoted by $$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2)),$$( E ( 2 ) ~ , g L ( λ 1 , λ 2 ) ) , where $$\lambda _1\ge \lambda _2>0.$$λ 1 ≥ λ 2 > 0 . It is one of three-dimensional unimodular Lie groups which are classified by Milnor. In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $$C^2$$C 2 -smooth surface in $$(\widetilde{E(2)},g_L(\lambda _1,\lambda _2))$$( E ( 2 ) ~ , g L ( λ 1 , λ 2 ) ) away from characteristic points and signed geodesic curvature for Euclidean $$C^2$$C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the universal covering group of Euclidean motion group E(2) with the general left-invariant metric.

The rototranslation group ℛ T is the group comprising rotations and translations of the Euclidean plane which is a 3-dimensional Lie group. In this paper, we use the Riemannian approximation scheme to compute sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in the rototranslation group away from characteristic points and signed geodesic curvature for Euclidean C 2 -smooth curves on surfaces. Based on these results, we obtain a Gauss–Bonnet theorem in the rototranslation group.

The relatively parallel adapted frame or Bishop frame is an alternative approach to define a moving frame that is well defined even when the curve has vanished second derivative, and it has been widely used in the areas of biology, engineering, and computer graphics. The main result of this paper is using the relatively parallel adapted frame for classification of singularity type of Bishop spherical Darboux image and Bishop dual which are deeply related with a space curve and making them visualized by computer.